The average of ten numbers is 27. If the smallest number is deleted from the list, the average of the remaining 9 numbers will be x and if the largest number is deleted, the average will be y. What is the average of the smallest and the largest numbers, if x + y = 10?a)225b)250c)275d)290Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Let the sum of all 10 numbers be S.
S = 10 × 27 = 270
Let the smallest number be s and the largest number be L.
So, S - s = 9x ... (i)
Also, S - L = 9y ... (ii)
Adding (i) and (ii), we get
2S - (s + L) = 9(x + y) = 9(x + y) = 9(10) = 90
Or 2S - 90 = (s + L)

Or 270 - 45 = 225

=

Thus, answer option (1) is correct.

Answer

Let's assume that the sum of the ten numbers is S. Since the average of the ten numbers is 27, we can write the equation:

S/10 = 27

From this equation, we can determine that the sum of the ten numbers is 270.

Let's also assume that the smallest number in the list is A and the largest number is B. We are asked to find the average of the smallest and largest numbers, which is (A + B)/2.

Now, let's consider the new scenario where the smallest number is deleted from the list. In this case, the sum of the remaining nine numbers would be S - A. The average of these nine numbers is given as x. We can write the equation:

(S - A)/9 = x

Similarly, in the scenario where the largest number is deleted, the sum of the remaining nine numbers would be S - B, and the average is given as y:

(S - B)/9 = y

We are also given that x - y = 10.

To solve this problem, we need to find the values of x, y, A, and B.

Let's solve the equations one by one:

1. S/10 = 27
S = 270

2. (S - A)/9 = x
(270 - A)/9 = x
270 - A = 9x
A = 270 - 9x

3. (S - B)/9 = y
(270 - B)/9 = y
270 - B = 9y
B = 270 - 9y

Now, let's substitute the values of A and B in the equation (A + B)/2:

[(270 - 9x) + (270 - 9y)]/2 = (540 - 9x - 9y)/2 = 270 - 4.5x - 4.5y

Since x - y = 10, we can substitute y = x - 10:

270 - 4.5x - 4.5(x - 10) = 270 - 4.5x - 4.5x + 45 = 315 - 9x

Now, we are given that x - y = 10, so x = y + 10. Substituting this in the equation:

315 - 9(y + 10) = 315 - 9y - 90 = 225 - 9y

Since x - y = 10, we can substitute x = y + 10:

225 - 9(y + 10) = 225 - 9y - 90 = 135 - 9y

Since we are given that x - y = 10, we can substitute y = x - 10:

135 - 9(x - 10) = 135 - 9x + 90 = 225 - 9x

Therefore, the average of the smallest and largest numbers is 225. Hence, option A is the correct answer.

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